On a spectral sequence for twisted cohomologies

Abstract

Let ($Ω^{\ast}(M), d$) be the de Rham cochain complex for a smooth compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$, there are a twisted de Rham cochain complex $(Ω^{\ast}(M), d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the filtration $F_p(Ω^{\ast}(M))=\bigoplus_{i\geq p}Ω^i(M)$ of $Ω^{\ast}(M)$, which converges to the twisted de Rham cohomology $H^*(M,H)$. We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.